Handbook of mathematics for engineers and scienteists part 103 pot

We will refer to solutions 15.5.1.1 of nonlinear equations 15.5.1.2 as generalized separable solutions.. Expressions of the form 15.5.1.1 are often used in applied and computational math

Linear separable equations of mathematical physics admit exact solutions in the form

w (x, y) = ϕ1(x)ψ1(y) + ϕ2(x)ψ2(y) + · · · + ϕ n (x)ψ n (y), (15.5.1.1)

where w i = ϕ i (x)ψ i (y) are particular solutions; the functions ϕ i (x), as well as the functions

ψ i (y), with different numbers i, are not related to one another.

Many nonlinear partial differential equations with quadratic or power nonlinearities,

f1(x)g1(y)Π1[w] + f2(x)g2(y)Π2[w] + · · · + f m (x)g m (y)Πm [w] =0, (15.5.1.2) also have exact solutions of the form (15.5.1.1) In (15.5.1.2), theΠi [w] are differential

forms that are the products of nonnegative integer powers of the function w and its partial derivatives ∂ x w , ∂ y w , ∂ xx w , ∂ xy w , ∂ yy w , ∂ xxx w, etc We will refer to solutions (15.5.1.1) of

nonlinear equations (15.5.1.2) as generalized separable solutions Unlike linear equations,

in nonlinear equations the functions ϕ i (x) with different subscripts i are usually related to one another [and to functions ψ j (y)] In general, the functions ϕ i (x) and ψ j (y) in (15.5.1.1)

are not known in advance and are to be identified Subsection 15.4.2 gives simple examples

of exact solutions of the form (15.5.1.1) with n =1and n =2(for ψ1 = ϕ2 =1) to some nonlinear equations

Note that most common of the generalized separable solutions are solutions of the special form

w (x, y) = ϕ(x)ψ(y) + χ(x);

the independent variables on the right-hand side can be swapped In the special case of

χ (x) =0, this is a multiplicative separable solution, and if ϕ(x) = 1, this is an additive separable solution

Remark Expressions of the form (15.5.1.1) are often used in applied and computational mathematics for constructing approximate solutions to differential equations by the Galerkin method (and its modifications).

15.5.1-2 General form of functional differential equations

In general, on substituting expression (15.5.1.1) into the differential equation (15.5.1.2), one arrives at a functional differential equation

Φ1(X)Ψ1(Y ) +Φ2(X)Ψ2(Y ) + · · · + Φ k (X)Ψk (Y ) =0 (15.5.1.3)

for the ϕ i (x) and ψ i (y) The functionals Φj (X) and Ψj (Y ) depend only on x and y,

respectively,

Φj (X)≡Φj x , ϕ1, ϕ 1, ϕ 1, , ϕ n , ϕ  n , ϕ  n

,

Ψj (Y )≡Ψj y , ψ1, ψ1 , ψ1 , , ψ n , ψ  n , ψ n 

Here, for simplicity, the formulas are written out for the case of a second-order equa-tion (15.5.1.2); for higher-order equaequa-tions, the right-hand sides of relaequa-tions (15.5.1.4) will

contain higher-order derivatives of ϕ i and ψ j

Subsections 15.5.3 and 15.5.4 outline two different methods for solving functional differential equations of the form (15.5.1.3)–(15.5.1.4)

Remark Unlike ordinary differential equations, equation (15.5.1.3)–(15.5.1.4) involves several functions (and their derivatives) with different arguments.

15.5.2 Simplified Scheme for Constructing Solutions Based on

Presetting One System of Coordinate Functions

15.5.2-1 Description of the simplified scheme

To construct exact solutions of equations (15.5.1.2) with quadratic or power nonlinearities

that do not depend explicitly on x (all f i constant), it is reasonable to use the following simplified approach As before, we seek solutions in the form of finite sums (15.5.1.1) We assume that the system of coordinate functions{ϕi (x)}is governed by linear differential equations with constant coefficients The most common solutions of such equations are of the forms

ϕ i (x) = x i ϕ i (x) = e λ i x, ϕ i (x) = sin(α i x), ϕ i (x) = cos(β i x) (15.5.2.1) Finite chains of these functions (in various combinations) can be used to search for separable

solutions (15.5.1.1), where the quantities λ i , α i , and β i are regarded as free parameters The other system of functions {ψi (y)} is determined by solving the nonlinear equations resulting from substituting (15.5.1.1) into the equation under consideration [or into equation (15.5.1.3)–(15.5.1.4)]

This simplified approach lacks the generality of the methods outlined in Subsections 15.5.3–15.5.4 However, specifying one of the systems of coordinate functions, {ϕi (x)}, simplifies the procedure of finding exact solutions substantially The drawback of this approach is that some solutions of the form (15.5.1.1) can be overlooked It is significant that the overwhelming majority of generalized separable solutions known to date, for partial differential equations with quadratic nonlinearities, are determined by coordinate

functions (15.5.2.1) (usually with n =2)

15.5.2-2 Examples of finding exact solutions of second- and third-order equations Below we consider specific examples that illustrate the application of the above simplified scheme to the construction of generalized separable solutions of second- and third-order nonlinear equations

Example 1 Consider a nonhomogeneous Monge–Amp`ere equation of the form



∂2w

∂x∂y

2 –∂

2w

∂x2

∂2w

We look for generalized separable solutions with the form

On substituting (15.5.2.3) into (15.5.2.2) and collecting terms, we obtain

[k2(ϕ  x)2– k(k –1)ϕϕ

xx ]y2k–2– k(k –1)ϕψ

xx y k–2– f (x) =0 (15.5.2.4)

This equation can be satisfied only if k =1or k =2.

First case If k =1, (15.5.2.4) reduces one equation

(ϕ  x)2– f (x) =0.

It has two solutions: ϕ(x) = 7 √

f (x) dx They generate two solutions of equation (15.5.2.2) in the form

(15.5.2.3):

w (x, y) = y

f (x) dx + ψ(x), where ψ(x) is an arbitrary function.

Second case If k =2, equating the functional coefficients of the different powers of y to zero, we obtain two equations:

2(ϕ

x)2– ϕϕ  xx= 0,

2ϕψ xx  – f (x) =0.

Their general solutions are given by

ϕ (x) = 1

C1x + C2, ψ (x) =

1 2

 x 0

(x – t)(C1t + C2)f (t) dt + C3x + C4.

Here, C1, C2, C3, and C4are arbitrary constants.

Table 15.4 lists generalized separable solutions of various nonhomogeneous Monge– Amp`ere equations of the form



∂2w

∂x∂y

2 – ∂

2w

∂x2

∂2w

Equations of this form are encountered in differential geometry, gas dynamics, and mete-orology

Example 2 Consider the third-order nonlinear equation

∂2w

∂x∂t+



∂w

∂x

2

– w ∂

2w

∂x2 = a ∂

3w

which is encountered in hydrodynamics.

We look for exact solutions of the form

On substituting (15.5.2.7) into (15.5.2.6), we have

ϕ  t – λϕψ = aλ2ϕ.

We now solve this equation for ψ and substitute the resulting expression into (15.5.2.7) to obtain a solution of

equation (15.5.2.6) in the form

w = ϕ(t)e λx+ 1

λ

ϕ  t (t)

ϕ (t) – aλ, where ϕ(t) is an arbitrary function and λ is an arbitrary constant.

15.5.3 Solution of Functional Differential Equations

by Differentiation

15.5.3-1 Description of the method

Below we describe a procedure for constructing solutions to functional differential equations

of the form (15.5.1.3)–(15.5.1.4) It involves three successive stages

1◦ Assume thatΨk0 We divide equation (15.5.1.3) byΨkand differentiate with respect

to y This results in a similar equation but with fewer terms:

2Φ1(X) 2Ψ1(Y ) + 2Φ2(X) 2Ψ2(Y ) + · · · + 2Φ k–1(X) 2Ψk–1(Y ) =0, 2Φj (X) =Φj (X), Ψ2j (Y ) = [Ψj (Y )/Ψk (Y )] 

y.

We repeat the above procedure (k –3) times more to obtain the separable two-term equation

$Φ1(X) $Ψ1(Y ) + $Φ2(X) $Ψ2(Y ) =0 (15.5.3.1)

TABLE 15.4

Exact solutions of a nonhomogeneous Monge–Amp`ere equation of the form (15.5.2.5); f (x), g(x), and h(x) are arbitrary functions; C1, C2, C3 , and β are arbitrary constants; a, b, k, and λ are some numbers (k≠ –1, –2)

2

4C1x

2 – 1

2C1

a (x – t)f (t) dt

x + C1



C2y2+ C3y+ C

2

4C2

 – 1

2C2

a (x – t)(t + C1)f (t) dt

3 f (x)y C1y2+ ϕ(x)y + ψ(x) (ϕ and ψ are expressed as quadratures)

x + C1 y

2+ ϕ(x)y + ψ(x) (ϕ and ψ are expressed as quadratures)

6C1

a (x – t)f (t) dt

(x + C1)2 –

1 6

a (x – t)(t + C1)2f (t) dt

7 f (x)y2 ϕ (x)y2+ ψ(x)y + χ(x) (ϕ, ψ, and χ are determined by ODEs)

12C1

a (x – t)f (t) dt

4

(x + C1) 3 – 1

12

a (x – t)(t + C1)3f (t) dt

10 f (x)y2+ g(x)y + h(x) ϕ (x)y2+ ψ(x)y + χ(x) (ϕ, ψ, and χ are determined by ODEs)

k+2

(k +1)(k + 2) –

1

C1

a (x – t)f (t) dt

k+2

(x + C1)k+1 – 1

(k +1)(k + 2)

a (x – t)(t + C1) k+1f (t) dt

13 f (x)y2k+2+ g(x)y k ϕ (x)y

k+2 – 1

(k +1)(k + 2)

a (x – t) g (t)

ϕ (t) dt (ϕ is determined by an ODE)

14 f (x)e λy C1e βx+λy– 1

C1λ2

a (x – t)e–βtf (t) dt

λy– 1

λ2

a (x – t) g (t)

ϕ (t) dt (ϕ is determined by an ODE)

a (x – t)f (t) dt – 1

C1

b (y – ξ)g(ξ) dξ λxy

Three cases must be considered

Nondegenerate case: |$Φ1(X)|+|$Φ2(X)|  0 and |Ψ1$ (Y )|+ |Ψ2$ (Y )|  0 Then the solutions of equation (15.5.3.1) are determined by the ordinary differential equations

$Φ1(X) + C $Φ2(X) =0, C $Ψ1(Y ) – $Ψ2(Y ) =0,

where C is an arbitrary constant The equations $Φ2=0and $Ψ1=0correspond to the limit

case C = ∞.

Two degenerate cases:

$Φ1(X)≡ 0, $Φ2(X)≡ 0 =⇒ $Ψ1, 2(Y ) are any functions;

$

Ψ1(Y )≡ 0, Ψ2$ (Y )≡ 0 =⇒ $Φ1,2(X) are any functions.

2◦ The solutions of the two-term equation (15.5.3.1) should be substituted into the original

functional differential equation (15.5.1.3) to “remove” redundant constants of integration [these arise because equation (15.5.3.1) is obtained from (15.5.1.3) by differentiation]

3◦ The caseΨk≡ 0should be treated separately (since we divided the equation byΨkat the first stage) Likewise, we have to study all other cases where the functionals by which the intermediate functional differential equations were divided vanish

Remark 1 The functional differential equation (15.5.1.3) happens to have no solutions.

Remark 2 At each subsequent stage, the number of terms in the functional differential equation can be

reduced by differentiation with respect to either y or x For example, we can assume at the first stage that

Φk 0 On dividing equation (15.5.1.3) by Φk and differentiating with respect to x, we again obtain a similar

equation that has fewer terms.

15.5.3-2 Examples of constructing exact generalized separable solutions

Below we consider specific examples illustrating the application of the above method of constructing exact generalized separable solutions of nonlinear equations

Example 1 The equations of a laminar boundary layer on a flat plate are reduced to a single third-order

nonlinear equation for the stream function (see Schlichting, 1981, and Loitsyanskiy, 1996):

∂w

∂y

∂2w

∂x∂y –∂w

∂x

∂2w

∂y2 = a ∂

3w

We look for generalized separable solutions to equation (15.5.3.2) in the form

On substituting (15.5.3.3) into (15.5.3.2) and canceling by ϕ, we arrive at the functional differential equation

ϕ  x [(ψ y )2– ψψ yy  ] – χ  x ψ  yy = aψ yyy  (15.5.3.4)

We differentiate (15.5.3.4) with respect to x to obtain

ϕ  xx [(ψ y )2– ψψ yy  ] = χ  xx ψ yy  (15.5.3.5)

Nondegenerate case On separating the variables in (15.5.3.5), we get

χ  xx = C1 ϕ  xx,

(ψ  y)2– ψψ yy  – C1ψ yy  = 0.

Integrating yields

ψ (y) = C4e λy – C1, ϕ (x) is any function, χ (x) = C1ϕ (x) + C2x + C3, (15.5.3.6)

where C1, , C4, and λ are constants of integration On substituting (15.5.3.6) into (15.5.3.4), we establish the relationship between constants to obtain C2= –aλ Ultimately, taking into account the aforesaid and formulas

(15.5.3.3) and (15.5.3.6), we arrive at a solution of equation (15.5.3.2) of the form (15.5.3.3):

w (x, y) = ϕ(x)e λy – aλx + C, where ϕ(x) is an arbitrary function and C and λ are arbitrary constants (C = C3, C4= 1).

Degenerate case It follows from (15.5.3.5) that

ϕ  xx= 0, χ  xx= 0, ψ (y) is any function. (15.5.3.7)

Integrating the first two equations in (15.5.3.7) twice yields

ϕ (x) = C1 x + C2, χ (x) = C3 x + C4, (15.5.3.8)

where C1, , C4are arbitrary constants.

Substituting (15.5.3.8) into (15.5.3.4), we arrive at an ordinary differential equation for ψ = ψ(y):

C1(ψ y )2– (C1ψ + C3)ψ yy  = aψ  yyy (15.5.3.9) Formulas (15.5.3.3) and (15.5.3.8) together with equation (15.5.3.9) determine an exact solution of equa-tion (15.5.3.2).

Example 2 The two-dimensional stationary equations of motion of a viscous incompressible fluid are

reduced to a single fourth-order nonlinear equation for the stream function (see Loitsyanskiy, 1996):

∂w

∂y

∂

∂x(Δw) – ∂w

∂x

∂

∂y(Δw) = aΔΔw, Δw = ∂2w

∂x2 +∂

2w

Here, a is the kinematic viscosity of the fluid and x, y are Cartesian coordinates.

We seek exact separable solutions of equation (15.5.3.10) in the form

Substituting (15.5.3.11) into (15.5.3.10) yields

g y  f xxx  – f x  g yyy  = af xxxx  + ag  yyyy (15.5.3.12)

Differentiating (15.5.3.12) with respect to x and y, we obtain

g yy f xxxx  – f xx  g yyyy  = 0 (15.5.3.13)

Nondegenerate case If f xx   0and g yy   0, we separate the variables in (15.5.3.13) to obtain the ordinary differential equations

which have different solutions depending on the value of the integration constant C.

1◦ Solutions of equations (15.5.3.14) and (15.5.3.15) for C =0:

f (x) = A1 + A2 x + A3 x2+ A4 x3,

g (y) = B1+ B2y + B3y2+ B4y3, (15.5.3.16)

where A k and B k are arbitrary constants (k = 1, 2, 3, 4) On substituting (15.5.3.16) into (15.5.3.12), we evaluate the integration constants Three cases are possible:

A4= B4= 0, A n , B nare any numbers (n =1, 2, 3);

A k= 0, B kare any numbers (k =1, 2, 3, 4);

B k= 0, A kare any numbers (k =1, 2, 3, 4).

The first two sets of constants determine two simple solutions (15.5.3.11) of equation (15.5.3.10):

w = C1x2+ C2x + C3y2+ C4y + C5,

w = C1 y3+ C2 y2+ C3 y + C4, where C1, , C5are arbitrary constants.

2◦ Solutions of equations (15.5.3.14) and (15.5.3.15) for C = λ2> 0:

f (x) = A1 + A2 x + A3 e λx + A4 e–λx,

g (y) = B1+ B2y + B3e λy + B4e–λy (15.5.3.17)

Substituting (15.5.3.17) into (15.5.3.12), dividing by λ3, and collecting terms, we obtain

A3(aλ – B2)e λx + A4(aλ + B2)e–λx+ B3(aλ + A2)e λy + B4(aλ – A2)e–λy= 0.

Equating the coefficients of the exponentials to zero, we find

A3= A4= B3= 0, A2= aλ (case 1),

A3= B3= 0, A2= aλ, B2 = –aλ (case 2),

A3= B4= 0, A2= –aλ, B2= –aλ (case 3).

(The other constants are arbitrary.) These sets of constants determine three solutions of the form (15.5.3.11) for equation (15.5.3.10):

w = C1e–λy + C2y + C3+ aλx,

w = C1e–λx+ aλx + C2e–λy– aλy + C3,

w = C1 e–λx– aλx + C2 e λy – aλy + C3,

where C1, C2, C3, and λ are arbitrary constants.

3◦ Solution of equations (15.5.3.14) and (15.5.3.15) for C = –λ2< 0:

f (x) = A1+ A2x + A3cos(λx) + A4sin(λx),

g (y) = B1 + B2 y + B3 cos(λy) + B4 sin(λy). (15.5.3.18) Substituting (15.5.3.18) into (15.5.3.12) does not yield new real solutions.

Degenerate cases If f xx  ≡ 0or g yy  ≡ 0, equation (15.5.3.13) becomes an identity for any g = g(y) or

f = f (x), respectively These cases should be treated separately from the nondegenerate case For example, if

f xx≡ 0, we have f (x) = Ax + B, where A and B are arbitrary numbers Substituting this f into (15.5.3.12), we

arrive at the equation –Ag yyy  = ag  yyyy Its general solution is given by g(y) = C1 exp(–Ay/a)+C2 y2+C3 y +C4.

Thus, we obtain another solution of the form (15.5.3.11) for equation (15.5.3.10):

w = C1 e–λy+ C2 y2+ C3 y + C4 + aλx (A = aλ, B =0).

15.5.4 Solution of Functional-Differential Equations by Splitting

15.5.4-1 Preliminary remarks Description of the method

As one reduces the number of terms in the functional differential equation (15.5.1.3)– (15.5.1.4) by differentiation, redundant constants of integration arise These constants must

be “removed” at the final stage Furthermore, the resulting equation can be of a higher order than the original equation To avoid these difficulties, it is convenient to reduce the solution

of the functional differential equation to the solution of a bilinear functional equation of a standard form and solution of a system of ordinary differential equations Thus, the original problem splits into two simpler problems Below we outline the basic stages of the splitting method

1◦ At the first stage, we treat equation (15.5.1.3) as a purely functional equation that

depends on two variables X and Y , whereΦn = Φn (X) andΨn = Ψn (Y ) are unknown

quantities (n =1, , k).

It can be shown* that the bilinear functional equation (15.5.1.3) has k –1 different solutions:

Φi (X) = C i,1Φm+1(X) + C i,2Φm+2(X) + · · · + C i,k–mΦk (X), i=1, , m;

Ψm+j (Y ) = –C1 ,jΨ1(Y ) – C2,jΨ2(Y ) – · · · – C m,jΨm (Y ), j=1, , k – m;

m=1,2, , k –1;

(15.5.4.1)

where C i,j are arbitrary constants The functions Φm+1(X), , Φk (X), Ψ1(Y ), ,

Ψm (Y ) on the right-hand sides of formulas (15.5.4.1) are defined arbitrarily It is apparent that for fixed m, solution (15.5.4.1) contains m(k – m) arbitrary constants.

* These solutions can be obtained by differentiation following the procedure outlined in Subsection 15.5.3, and by induction Another simple method for finding solutions is described in Paragraph 15.5.4-2, Item 3◦.

of the functional differential equation to the solution of a bilinear functional equation of a standard form and solution of a system of ordinary differential equations... Ultimately, taking into account the aforesaid and formulas

(15.5.3.3) and (15.5.3.6), we arrive at a solution of equation (15.5.3.2) of the form (15.5.3.3):

w... 15.4

Exact solutions of a nonhomogeneous Monge–Amp`ere equation of the form (15.5.2.5); f (x), g(x), and h(x) are arbitrary functions; C1, C2, C3 , and β are arbitrary constants; a, b, k, and λ are some